Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli

• An integral representation for viscoelastic attenuation and dispersion is obtained.
• The attenuation and dispersion function satisfy a Kramers–Kronig dispersion relation.
• Low-frequency behavior of viscoelastic attenuation is consistent with experiment.
• Asymptotic growth of attenuation in the high-frequency range is sublinear.
• Minimum phase properties of viscoelastic Green’s functions are examined.

It is shown that viscoelastic wave dispersion and attenuation in a viscoelastic medium with a completely monotonic relaxation modulus is completely characterized by the phase speed and the dispersion–attenuation spectral measure. The dispersion and attenuation functions are expressed in terms of a single dispersion–attenuation spectral measure. An alternative expression of the mutual dependence of the dispersion and attenuation functions, known as the Kramers–Kronig dispersion relation, is also derived from the theory. The minimum phase aspect of the filters involved in Green’s function is another consequence of the theory. Explicit integral expressions for the attenuation and dispersion functions are obtained for a few analytical relaxation models.